Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

Table of Notes

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          <head xml:id="echoid-head300" xml:space="preserve">THEOR. XXXII. PROP. LI.</head>
          <p>
            <s xml:id="echoid-s6933" xml:space="preserve">MINIMA portionum eiuſdem anguli, vel cuiuslibet coni-ſectio-
              <lb/>
            nis, quarum altitudines ſint equales, eſt ea, cuius diameter ſit ſegmẽ-
              <lb/>
            tum maioris axis: </s>
            <s xml:id="echoid-s6934" xml:space="preserve">in Ellipſi verò MAXIMA eſt, cuius diameter ſit
              <lb/>
            ſegmentum minoris axis.</s>
            <s xml:id="echoid-s6935" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6936" xml:space="preserve">ESto A B C, in prima figura, angulus rectilineus, vel in ſecunda, Parabole,
              <lb/>
            aut Hyperbole, ſiue in tertia Ellipſis, quarum axes ſint B D, at in Ellipſi
              <lb/>
            axis maior ſit B D N, minor L K; </s>
            <s xml:id="echoid-s6937" xml:space="preserve">centrum E, atque axi B D in quauis figura
              <lb/>
            applicata ſit quælibet A D C. </s>
            <s xml:id="echoid-s6938" xml:space="preserve">Dico portionem A B C, quæ in tertia figura
              <lb/>
            ſit, vel maior, vel minor ſemi-Ellipſi, eſſe _MINIMAM_ omnium portionum
              <lb/>
            eiuſdem anguli, vel coni-ſectionis, quarum altitudines ſint æquales ipſi B D.</s>
            <s xml:id="echoid-s6939" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6940" xml:space="preserve">Deſcripta. </s>
            <s xml:id="echoid-s6941" xml:space="preserve">n. </s>
            <s xml:id="echoid-s6942" xml:space="preserve">per D, vel Hyperbola
              <lb/>
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            in prima figura, cuius aſymptoti ſint B
              <lb/>
            A, B C; </s>
            <s xml:id="echoid-s6943" xml:space="preserve">vel in reliquis figuris, deſcri-
              <lb/>
            pta eiuſdem nominis coni-ſectione ſi-
              <lb/>
            mili, & </s>
            <s xml:id="echoid-s6944" xml:space="preserve">concentrica F D G, que rectam
              <lb/>
            A D C continget in D; </s>
            <s xml:id="echoid-s6945" xml:space="preserve">ſumatur in in-
              <lb/>
            teriori ſectione quodlibet aliud punctũ
              <lb/>
            F, ad quod ſit contingens H F I exte-
              <lb/>
            riori occurrens in H, I, atque portionẽ
              <lb/>
            abſcindens H O I, cuius diameter ſit
              <lb/>
            O F, altitudo verò ſit O P.</s>
            <s xml:id="echoid-s6946" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6947" xml:space="preserve">Itaque cum portio H O I equalis
              <note symbol="a" position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve">45. h.</note>
            portioni A B C eiuſdem ſectionis, erit
              <lb/>
            reciprocè baſis H I ad baſim A C, vt
              <lb/>
            altitudo B D ad altitudinem O P, ſed
              <lb/>
            eſt H I maior A C, cum A C ſit om-
              <lb/>
            nium contingentium _MINIMA_,
              <note symbol="b" position="right" xlink:label="note-0251-02" xlink:href="note-0251-02a" xml:space="preserve">47. h.</note>
            & </s>
            <s xml:id="echoid-s6948" xml:space="preserve">B D erit maior O P: </s>
            <s xml:id="echoid-s6949" xml:space="preserve">producatur ergo
              <lb/>
            O P, & </s>
            <s xml:id="echoid-s6950" xml:space="preserve">ſumatur O Q ipſi B D æqualis,
              <lb/>
            appliceturque S Q R contingenti H I
              <lb/>
            æquidiſtans: </s>
            <s xml:id="echoid-s6951" xml:space="preserve">eruntque portiones S O R, A B C æqualium altitudinum, ſed eſt
              <lb/>
            portio H O I minor S O R, pars ſuo toto, ergo, & </s>
            <s xml:id="echoid-s6952" xml:space="preserve">portio A B C, quæ ipſi H O I
              <lb/>
            eſt æqualis, minor erit portione S O R, & </s>
            <s xml:id="echoid-s6953" xml:space="preserve">hoc ſemper, &</s>
            <s xml:id="echoid-s6954" xml:space="preserve">c. </s>
            <s xml:id="echoid-s6955" xml:space="preserve">Vnde portio A B
              <lb/>
            C eſt _MINIMA_ portionum eiuſdem anguli, vel coni-ſectionis, & </s>
            <s xml:id="echoid-s6956" xml:space="preserve">æqualium
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            altitudinum. </s>
            <s xml:id="echoid-s6957" xml:space="preserve">Quod primò erat, &</s>
            <s xml:id="echoid-s6958" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6959" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6960" xml:space="preserve">Ampliùs in tertia figura eſto recta V K T minori axi L M ordinatim appli-
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            cata. </s>
            <s xml:id="echoid-s6961" xml:space="preserve">Dico portionem V M T (quæ ſit vel maior, vel minor ſemi-Ellipſi) cu-
              <lb/>
            ius diameter, vel altitudo eſt M K, eſſe _MAXIMAM_ portionum omnium, qua-
              <lb/>
            rum altitudines ipſi M K ſint æquales.</s>
            <s xml:id="echoid-s6962" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s6963" xml:space="preserve">Deſcripta enim per K Ellipſi K D G ſimili, & </s>
            <s xml:id="echoid-s6964" xml:space="preserve">concentrica datæ A B C N,
              <lb/>
            quæ rectam V K T continget in K, ſumptoque in eius peripheria quocunque
              <lb/>
            puncto F, ducatur contingens H F I exteriori ſectioni occurrens in H, I, de-
              <lb/>
            que ipſa abſcindens portionem I X H, cuius diameter ſit F X, altitudo verò
              <lb/>
            ſit X Z.</s>
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